A299427 Square table where T(n,k) = binomial(n*(n+k), k) * n/(n+k), for n>=1, k>=0, as read by antidiagonals.

1, 1, 1, 4, 1, 1, 9, 14, 1, 1, 16, 63, 48, 1, 1, 25, 184, 408, 165, 1, 1, 36, 425, 1872, 2565, 572, 1, 1, 49, 846, 6175, 17980, 15939, 2002, 1, 1, 64, 1519, 16536, 82775, 167552, 98670, 7072, 1, 1, 81, 2528, 38318, 292581, 1059380, 1535352, 610740, 25194, 1, 1, 100, 3969, 79808, 861175, 4874688, 13177125, 13934752, 3786588, 90440, 1

Offset: 0

Author

Paul D. Hanna, Feb 19 2018

Keywords

• nonn
• tabl

Examples

			This table begins:
n=1: [1,  1,    1,      1,       1,         1,          1, ...];
n=2: [1,  4,   14,     48,     165,       572,       2002, ...];
n=3: [1,  9,   63,    408,    2565,     15939,      98670, ...];
n=4: [1, 16,  184,   1872,   17980,    167552,    1535352, ...];
n=5: [1, 25,  425,   6175,   82775,   1059380,   13177125, ...];
n=6: [1, 36,  846,  16536,  292581,   4874688,   78119454, ...];
n=7: [1, 49, 1519,  38318,  861175,  18008676,  358919022, ...];
n=8: [1, 64, 2528,  79808, 2214640,  56592320, 1367090208, ...];
n=9: [1, 81, 3969, 153117, 5132565, 157000275, 4507103601, ...];
...
Row generating functions R(x,n)^(n^2) begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...
R(x,2)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + 572*x^5  + ...
R(x,3)^9 = 1 + 9*x + 63*x^2 + 408*x^3 + 2565*x^4 + 15939*x^5 + ...
R(x,4)^16 = 1 + 16*x + 184*x^2 + 1872*x^3 + 17980*x^4 + 167552*x^5 + ...
R(x,5)^25 = 1 + 25*x + 425*x^2 + 6175*x^3 + 82775*x^4 + 1059380*x^5 + ...
R(x,6)^36 = 1 + 36*x + 846*x^2 + 16536*x^3 + 292581*x^4 + 4874688*x^5 + ...
...
Related series R(x,n) = 1 + x*R(x,n)^n begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
...
where R(x,n)^m = Sum_{k>=0} C(m + n*k, k) * m/(m + n*k) * x^k.
...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1079 of rows 1..45 as a flattened square table read by antidiagonals.

Crossrefs

Cf. A299044 (antidiagonal sums), A299428 (diagonal), A299429.

Programs

PARI

{T(n,k) = binomial(n*(n+k), k) * n/(n+k) }
/* Print as a square table of first 9 rows */
for(n=1,9,print1("n="n": [",); for(k=0,8,print1(T(n,k),", ")); print1("...];");print(""))
/* Print as a Flattened table read by antidiagonals */
for(n=1,10,for(k=0,n,print1(T(n-k+1,k),", ")))

Formula

G.f. for row n: R(x,n)^(n^2) = Sum_{k>=0} C(n*(n+k), k) * n/(n+k) * x^k, where R(x,n) = 1 + x*R(x,n)^n.